**Puzzle 75.**
>Suppose \$$(Y, \le_Y) \$$ is a preorder, \$$X\$$ is a set and \$$f : X \to Y\$$ is any function. Define a relation \$$\le_X\$$ on \$$X\$$ by

>$x \le_X x' \textrm{ if and only if } f(x) \le_Y f(x') .$

>Show that \$$(X, \le_X ) \$$ is a preorder.

Our relation turns \$$f \$$ into a *monotone map*. For all \$$x \in X \$$ we have \$$f(x) \le_Y f(x)\$$ and thus \$$x \le_X x\$$ satisfying reflexivity. Similarly, for all \$$x,y,z \in X \$$, \$$f(x) \le_Y f(y)\$$ and \$$f(y) \le_Y f(z)\$$ implies \$$f(x) \le_Y f(z)\$$ and thus \$$x \le_X y\$$ and \$$y \le_X z\$$ implies \$$x \le_X z\$$ satisfying transitivity. This gives us a preorder on X.

**Puzzle 76.**
>Now suppose \$$(Y, \le_Y) \$$ is a poset. Under what conditions on \$$f\$$ can we conclude that \$$(X, \le_X ) \$$ defined as above is a poset?

Since we don't want to induce any equivalent elements in \$$X\$$, \$$f \$$ must be injective.

**Puzzle 77.**
>Now suppose that \$$(Y, \le_Y, \otimes_Y, 1_Y) \$$ is a monoidal preorder, and \$$(X,\otimes_X,1_X ) \$$ is a monoid. Define \$$\le_X\$$ as above. Under what conditions on \$$f\$$ can we conclude that \$$(X,\le_X\otimes_X,1_X) \$$ is a monoidal preorder?

We need to assure that our induced preorder structure is compatible with our monoidal structure. To this end we require our *monotone map* \$$f \$$ to be a *monoidal monotone* for which \$$1_Y \le_Y f(1_X) \$$ and \$$f(x) \otimes_Y f(y) \le_Y f(x \otimes_X y) \$$

Regarding Puzzle 71, does this mean we simply need to find injective *monoidal monotones* to other commutative monoidal posets (e.g \$$(\mathbb{R}, \le, +, 0 )\$$) or do we need stricter requirements to preserve the commutative sturcture (e.g \$$1_Y = f(1_X) \$$ and \$$f(x) \otimes_Y f(y) = f(x \otimes_X y) \$$)?

I'm off to bed, so maybe someone else can continue my thought process...